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Thanks, hectictar...I forgot that a, b, c had to be integers......

But...if we did that....we would just end up with what we started with.....

Besides   1 / √2    is just a number.......there's no "voodoo" associated with having a root in the denominator

9x² - 18x - 720  =  0       Divide through by  9 .

x² - 2x - 80  =  0             Add  80  to both sides.

x² - 2x  =  80                  Add  (2/2)²  , which is  1 ,  to both sides of the equation.

x² - 2x + 1  =  80 + 1      Factor the left side as a perfect square trinomial.

(x - 1)(x - 1)  =  81

(x - 1)²  =  81

Now we have it in the form  (x + r)²  =  s  , and we can see that  s = 81 .

Happy belated birthday!!!

The aops answer says 1/2. May someone explain it to me?

I agree with you CPhill, but isn't a fraction not allowed to have a square root in the denominator?

I think we are supposed to multiply both the numerator and denominator by the square root of 2. 

√2/2 is the final answer. 

(x - $\frac47$)²  = $\frac{80}{49}$

On this one...to get  a ,  b , and  c  as integers, we can multiply both sides by  7²

7² * (x - $\frac47$)²  = 7² * $\frac{80}{49}$

( 7 * (x - $\frac47$) )²  =  49 * $\frac{80}{49}$

(7x - 4)²  =  80

And...    7 + (-4) + 80  =  83      

Note that 2  = √2 * √2

So

√2 / 2  =  √2 /   [√2 * √2 ]   =    1 / √2

Starting with 0,  78987  is the 889th palindrome 

[BTW...889 is known as the palindrome's rank ]

Starting with 1, it is the 888th palindrome

This can be verified with the calculator here : 

http://rhyscitlema.com/algorithms/generating-palindromic-numbers/

[ The calculator is about half-way down the page.....there are also other intems of interest related to palindromes on this page ]

A procedure is desribed for finding the nth palindrome, but I don't see one that tells us how to determine the rank of any particular palindrome  [ although I did not look at the page with a fine tooth comb]

Maybe heureka knows of a proceedure to produce this???

y  =  60.18x + 335.30

This is a linear equation, so if we just have two points we can draw a straight line through them.

When  x = 0 ,

y  =  60.18(0) + 335.30  =  335.30

So we know the graph contains the point  (0, 335.3)

When  y = 0,

0  =  60.18x + 335.30   →   x  =  -335.30 / 60.18  ≈  -5.572

So we know the graph contains the point  (-5.572, 0)

Plot the points  (0,335.3)  and  (-5.572, 0), and draw a straight line through them,

s/4 - 2t  =  2     Add  2t  to both sides of this equation.

s/4  =  2 + 2t    Multiply through by  4 .

s  =  8 + 8t

3t - 6s  =  9     Since  s = 8 + 8t , we can substitute  8 + 8t  in for  s .

3t - 6(8 + 8t)  =  9        Distribute the  -6 .

3t - 48 - 48t  =  9        Add  48  to both sides.

3t - 48t  =  9 + 48       Combine like terms.

-45t  =  57        Divide both sides by  -45 .

t  =  -57/45       Reduce the numerator and denominator by  3 .

t  =  -19/15

To find  s , plug this value for  t  into one of the equations. I pick the equation  s = 8 + 8t .

s  =  8 + 8(-19/15)

s  =  8 -  152/15

s  =  -32/15

Using a proportion we have

125 / 2.5  =  Distance / 4.2      multipl both sides by 4.2

125 (4.2) / 2.5  =  Distance  =  210 m

Thank you so much!

Note that 74  km  =   74000 m  =  7400000 cm

So...it will be    7400000 / 2000000  =  37/10  cm  = 3.7 cm on the map

-5x^2+20x-17    divide through by - 5

-5 ( x^2 - 4x + 17/5)

Take 1/2 of 4  =  2.......square it = 4.....add it and subtract it

-5 ( x^2 - 4x + 4 + 17/5 - 4)

Factor the first three terms and simplify

-5 [ (x -2)^2  +  17/5 - 20/5 ] 

-5 [ (x - 2)^2  - 3/5]

-5( x - 2)^2  + 3

a = -5   d = -2    and  e = 3

So

a + d + e  =   -5 + (-2) + 3  =   -7 + 3  =  - 4 

Thanks this really helped!

49 x^2+ 56x- 64 = 0       divide through by 49

x^2 +  56/49 - 64/49  = 0

x^2 -  8/7 - 64/49  = 0       add 64/49 to both sides

x^2 - 8/7  =   64/49

Take 1/2 of 8/7 = 8/14= 4/7......square it  =  16/49   add it to both sides

x^2 - 8/7 + 16/49  =   64/49 + 16/49

x^2 - 8/7 +  16/49  =  80 / 49         factor the left side

(x - 4/7)^2  = 80/49    

So     a = 1, b = -4/7  and c = 80/49

And   a + b + c  =  1 - 4/7 + 80/49 =   [ 49 - 28 + 80] / 49  =  101 / 49

Thanks really but I'm not sure I understand the answer there... Can you expound please? thanks so much!

Follow the procedure shown at: https://web2.0calc.com/questions/if-we-express-3x-2-6x-2-in-the-form-a-x-h-2-k-what-is-a-h-k

See https://web2.0calc.com/questions/if-we-express-3x-2-6x-2-in-the-form-a-x-h-2-k-what-is-a-h-k#r4

Well, we have:

coeff of x^2:   a = 3

coeff of x:      -2ah = -6  → -2*3*h = -6  → 6h = 6  →  h = 1

constant;      ah^2 + k = -2 →  3*(1^2) + k = -2  →  3 + k = -2  →  k = -5

Hence  a + h + k = 3 + 1 - 5  or  a + h + k = -1

Please could someone respond?

supermanaccz:

No, this is a TVM formula(Time Value of Money) and he/she wants to find the compound interest rate.

However, they have written the formula wrong!

12,500 =500(1 + r)^5    divide both sides by 500

25 =(1+r)^5                      Take the 5th root of both sides

(1+r) =25^(1/5)

(1+r) =1.90365                Subtract 1 from sides

r =0.90365 x 100

r=90.365% which is the interest rate. However, it is so high, there is almost certainly something wrong with the way the formula is written. I suspect that it is something like this:

12,500 =500(1 + r/12)^(5*12)

12,500 =500(1 + r/12)^60, solve for r

r =66.14% - interest rate compounded monthly. Even this interest rate is way too high !!!.

Note to the questioner: Why don't you tell us what you are trying to calculate by stating the question as it is written in your book or assignment paper. Then will show you how to calculate the interest rate accurately.

The first step is that we can clear unnecessary numbers: 

12,500 = 5000 ( 1 + r ) ( 5 )

Then we divide both sides by 12,500

1 = 2 ( 1 + r )

Now that we got rid of the big numbers, we can distribute the parentheses. 

1 = 2 + 2r 

Subtract 2 from both sides:

2r = -1

Divide 2 from both sides: 

r = -1/2